Nonlinear model predictive control (NMPC) strategies based on linearization about predicted system trajectories enable the online NMPC optimization to be performed by a sequence of convex optimization problems. The approach relies on bounds on linearization errors in order to ensure constraint satisfaction and convergence of the performance index, both during the optimization at each sampling instant and along closed loop system trajectories. This technical note proposes bounds based on robust tubes constructed around predicted trajectories. To ensure local optimality, the bounds are non-conservative for the case of zero linearization error, which requires the tube cross sections to vary along predicted trajectories. The feasibility, stability and convergence properties of the algorithm are established without the need for predictions to satisfy local optimality criteria. The strategy is illustrated by numerical examples.